The.Algorithm.Design.Manual.Springer-Verlag.1998

Randomization
for i = low+1 to high
swap(A[low],A[leftwall])
if (A[i] < pivot) then
leftwall = leftwall+1
swap(A[i],A[leftwall])
Mergesort ran in time because we split the keys into two equal halves, sorted them recursively, and
then merged the halves in linear time. Thus whenever our pivot element is near the center of the sorted
array (i.e. the pivot is close to the median element), we get a good split and realize the same performance as
mergesort. Such good pivot elements will often be the case. After all, half the elements lie closer to the
middle than one of the ends. On average, Quicksort runs in time. If we are extremely unlucky and
our randomly selected elements always are among the largest or smallest element in the array, Quicksort
turn into selection sort and runs in . However, the odds against this are vanishingly small.
Randomization is a powerful, general tool to improve algorithms with bad worst-case but good averagecase
complexity. The worst case examples still exist, but they depend only upon how unlucky we are, not
on the order that the input data is presented to us. For randomly chosen pivots, we can say that
``Randomized quicksort runs in time on any input, with high probability.''
If instead, we used some deterministic rule for selecting pivots (like always selecting A[(low+high)/2] as
pivot), we can make no claim stronger than
``Quicksort runs in time if you give me random input data, with high probability.''
Randomization can also be used to drive search techniques such as simulated annealing, which are
discussed in Section .
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